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PHYSICS GRADUATE SCHOOL QUALIFYING EXAMINATION Partl INSTRUCTIONS: Work all problems. This is a closed book examination. Start each problem on a new pack of yellow paper and use only one side of each sheet. All problems carry the same weight. Write your student number on the upper right-hand corner of each answer sheet. 1. Two blocks are positioned as shown in the accompanying figure. The masses of the blocks are m and M, as shown. The coefficient of static friction between the two blocks is Jls, and the coefficient of sliding {kinetic) friction between the block of mass M and the horizontal surface is ~K. What constant horizontal force must be applied just to keep the block of mass m from sliding down the interface between the blocks? 3. A spherical capacitor consists of an outer conducting sphere of fixed radius band a concentric inner one of adjustable radius a. The space between the spheres is filled with air, which has a breakdown electric field strength Eo. a. Determine the greatest possible potential difference between the spheres. be Determine capacitor the greatest possible electrostatic energy stored in the 0 I 4. Two single-tum coaxial coils of radius r, each carrying a cuuent I in the same direction are separated by a distance d along their axis. Find the magnetic field on the axis. . : !d . . . 6. In traveling region wave 'P=A exp (iklX) I (V=O) an electron moving to the right is I where kl=(2mE/n2)1/2. If E > Vo find the wave reflected by the potential step and the transmitted wave into region II (V= V 0). What percentage of electrons of energy E > V0 is reflected and what percentage is transmitted? 7. A simple pendulum of length .e and bob mass m is suspended from a height H (>.e) above a grounded flat conducting plate. The bob is given a net charge q. Determine the frequency of small oscillations of the pendulum. IT 8. Two long conducting cylindrical tubes of radii a and b have a coaxial suspension which permits each cylinder to rotate independently and also provides electrical connections. The top ends of the tubes are closed by conducting disks. The system is suspended in a uniform magnetic field B which is parallel to the axis. Initially the B cylinders are at rest and uncharged. A potential difference is now applied through the coaxial suspension so that the cylindrical surface of the inner cylinder has a charge +Q and the outer cylinder has end plates, approximation. -Q. (Neglect i.e. we use ) a. Calculate the angular charging process. (Hint: Assume any charge on the a long cylinder Ir between momentum of each cylinder as a result of the The current flows in the top disks are radial. r=O and a or b.) b. Calculate the angular momentum of the combined electric and magnetic fields between the cylinders. Show that the total angular momentum of the entire system is still zero. (Hint: The linear momentum density of the combined E and B fields is p = EOE x B.) B .e PHYSICS GRADUATE SCHOOL Part QUALIFYING EXAMINATION II INSTRUCTIONS: Work all problems. This is a closed book examination. Start each problem on a new pack of yellow paper and use only one side of each sheet. All problems carry the same weight. Write your student number on the upper right-hand corner of each answer sheet. a. Set up equation b. Show the Lagrangian, and derive the of motion. that for small displacements from equilibrium, the sphere executes harmonic motion. Find the period. simple b. Find the torque exerted by the axle on the rod with respect to O at the instant shown. z Ce Determine the electric field in the plane of the ring near the center , ieee for sma11 r e 4. In a medium with a finite conductivity 0', an electric field is generated at time t = 0 such that E(t = 0) = Eo. Starting from Maxwell's equations, derive the time-evolution of this field. You may assume that B = 0. What is the dielectric 5. Consider relaxation the reaction time 'tdie in terms of the dielectric in which a photon collides permitivity with a proton E at rest to produce a neutral 7t meson in addition to the proton in the final state. Letting Mo be the proton rest mass and mo the pion rest mass find the minimum proceed. photon energy required for the reaction (y + p ~ p + 7rQ) to 6. A particle of mass m experiences where o(x) is a Dirac Schrodinger equation, a and n. an attractive potential of V(x)=-aO(x) delta function. Starting from the one-dimensional derive the energy of the bound state in terms of m, Use the fact that the fIrst discontinuous at the origin 7. Consider a thin circular a. Find the magnetic derivative of the wavefunction because of the delta function there. disk of radius R moment M of the system. b. Find the angular momentum L of the system and the ratio MIL. is 8. A solid copper ring (torus) minor radius a, and major b, such that a«b. of inertia about M~/2 has radius -HO The moment a diameter is for the ring, and its volume a 00 , .ty 6- is 27t2a2b. It rotates about a diameter without friction. There is a uniform magnetic field B perpendicular decrease to l/e of its original heating. ~ value ~ to the rotation assuming -axis. Assume all loss is due to resistive